Infinitesimal Gunk
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Infinitesimal Gunk Lu Chen1 Received: 6 March 2019 / Accepted: 3 January 2020 / © Springer Nature B.V. 2020
Abstract In this paper, I advance an original view of the structure of space called Infinitesimal Gunk. This view says that every region of space can be further divided and some regions have infinitesimal size, where infinitesimals are understood in the framework of Robinson’s (1966) nonstandard analysis. This view, I argue, provides a novel reply to the inconsistency arguments proposed by Arntzenius (2008) and Russell (2008), which have troubled a more familiar gunky approach. Moreover, it has important advantages over the alternative views these authors suggested. Unlike Arntzenius’s proposal, it does not introduce regions with no interior. It also has a much richer measure theory than Russell’s proposal and does not retreat to mere finite additivity. Keywords Continuum · Contact puzzle · Gunky space · Infinitesimals · Nonstandard analysis · Region-based topology · Nonstandard measure theory · Hyperfinite additivity
1 Is Space Pointy? Consider the space you occupy. Does it have ultimate parts? According to the standard view, the answer is yes: space is composed of uncountably many unextended points.1 Although standard, this view leads to many counterintuitive results. For example, intuitively, the size of a region should be the sum of the sizes of its disjoint parts.2 But according to the standard view, the points have zero size. Thus they cannot add up to a finite size, because zeros always add up to zero. For another example, according to the standard view, every region of space (except the whole space) has a boundary, and a closed region includes its boundary. 1 “Space” can be understood as physical space or (mathematical) geometric space: the discussions in this paper do not turn on the differences between them. Many considerations also apply to time or spacetime. 2 This is one of the intuitions behind Zeno’s paradox of measure. See Skyrms [14] and Butterfield [2].
Lu Chen
[email protected] 1
Philosophy Department, University of Massachusetts, Amherst, South College 305, Amherst, MA, USA
L. Chen
Now, suppose that two rigid bodies which occupy closed regions come into perfect contact: there is no gap between them. Under the standard view, we cannot put two closed regions side by side without overlapping and without leaving a gap between them. Thus, to be in perfect contact, the two closed regions must overlap on their boundaries. But the bodies are rigid and impenetrable, so they should not occupy overlapping regions. Therefore, if the standard view is true, two rigid bodies that occupy closed regions cannot come into perfect contact. But perfect contact is intuitively possible. This is called “the contact puzzle.” (See Zimmerman [19], Arntzenius [1] and Russell [11]) Due to these problems, a gunky conception of space has been proposed, according to which space cannot be broken down into ultimate parts. That is, every part of space can be further divided, and extensionless points do not exist. Such
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